Chapter 10, Problem

Analogies exist between rotational and translational physical quantities. Identity the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, "near momentum, Impulse.

To determine

To identify the rotational terms analogous to given translational physical quantities.

Answer

In rotational motion angular acceleration, torque, moment of inertia, rotational kinetic energy, angular momentum and angular impulse play the same role as acceleration, force, mass, translational kinetic energy, linear momentum and impulse respectively in linear motion.

Explanation of Solution

Given info:

The translational physical quantities are:

1. Acceleration
2. Force
3. Mass
4. Work
5. Translational kinetic energy
6. Linear momentum
7. Impulse

Identifying the quantities associated with translational motion and their analogues in rotational motion.

1. Acceleration

The analogue for linear acceleration $\left(a\right)$ in rotational motion is the angular acceleration $\left(\alpha \right)$.

The S.I. unit of linear acceleration is ${\text{ms}}^{\text{-2}}$ whereas the S.I. unit of angular acceleration is ${\text{rads}}^{\text{-2}}$ .

2. Force

Force $\left(F\right)$ is the product of mass and the linear acceleration. Torque $\left(\tau \right)$ is the analogue for force in rotational motion. Torque is the product of moment of inertia and angular acceleration.

The S.I. unit of Force is $\text{Newton}$ whereas the S.I. unit of torque is $\text{Newton-meter}$ .

3. Mass

In rotational motion moment of inertia $\left(I\right)$ is the analogue to the mass $\left(m\right)$ in linear motion.

The moment of inertia is the product of mass of the system of rotating particle and square of the perpendicular distance of particle form rotational axis. Mathematically $I=m{R}^2$.

The S.I. unit of mass is kilogram denoted by $\text{kg}$ and unit of moment of inertia is kilogram-meter-per-second -square denoted by $\text{kg}{\text{m}}^2$ .

4. Work:

In translational motion, work done is given by $\overrightarrow{F}.\overrightarrow{d}$ Where, $\overrightarrow{F}$ is the force and $\overrightarrow{d}$ is the displacement. In rotational motion about a fixed axis work done is the product of torque $\left(\overrightarrow{\tau }\right)$ and the angular displacement $\left(\overrightarrow{\theta }\right)$ represented by $\overrightarrow{\tau }.\overrightarrow{\theta }$ . The unit of work done in both

translational and rotational motion is the same i.e. Newton-meter.

5. Translational kinetic energy:

The translational depend upon the mass $\left(m\right)$ and the velocity $\left(v\right)$ of the object. Expressed as,

  $K.E{.}_{tran}=\frac{1}{2}m{v}^2$

In rotational motion rotational kinetic energy is considered. It is represented as,

  $K.E{.}_{rot}=\frac{1}{2}I{\omega }^2$

Here, $I$ is the moment of inertia, $\omega$ is the angular velocity.

The S.I. unit of kinetic energy in both translational and rotational motion is the same i.e. joule.

6. Linear momentum

Angular momentum is analogue to the linear momentum in rotational motion. As linear momentum $\left(p\right)$ is the product of mass $\left(m\right)$ and the linear velocity $\left(v\right)$ , similarly angular momentum $\left(L\right)$ is the product of its moment of inertia $\left(I\right)$ and its angular velocity $\left(\omega \right)$.

  $L=I\omega$

The S.I. unit of linear momentum is the kilogram meter per second but angular momentum is kilogram meter squared per second.

7. Impulse

Impulse $\left(I\right)$ is the integral of a force $\left(F\right)$ over the time interval $\left(t\right)$ . The rotational analog to translational impulse, is angular impulse. It is expressed as,

  $J=\tau t$

Here, $\tau$ is the torque and $t$ is the time interval.

The S.I. unit of linear impulse is Newton-meter whereas angular impulse is Newton-meter-second.

Conclusion:

Thus, different physical quantities associated with linear motion and their analogues in rotational motion. In rotational motion angular velocity, angular displacement, moment of insertia and torque play the same role as linear velocity, displacement, mass and force respectively in linear motion. With these basic analogue quantities, we are able to identify the other analogues quantities.